Toniolo and Linder, Equation (13)

Percentage Accurate: 50.5% → 59.4%

Time: 8.3s

Alternatives: 11

Speedup: 0.4×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (sqrt
  (*
   (* (* 2.0 n) U)
   (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt((((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)))));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)))));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 59.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;-1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (sqrt
          (*
           (* (* 2.0 n) U)
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_1 0.0)
     (* -1.0 (* (* (sqrt U) (sqrt (* n t))) (* -1.0 (sqrt 2.0))))
     (if (<= t_1 INFINITY)
       (sqrt
        (*
         (-
          (fma -2.0 (* l (/ l Om)) t)
          (* n (* (* (/ l Om) (/ l Om)) (- U U*))))
         (* (+ n n) U)))
       (sqrt
        (*
         (* (* (* (* l l) n) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = sqrt((((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = -1.0 * ((sqrt(U) * sqrt((n * t))) * (-1.0 * sqrt(2.0)));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = sqrt(((fma(-2.0, (l * (l / Om)), t) - (n * (((l / Om) * (l / Om)) * (U - U_42_)))) * ((n + n) * U)));
	} else {
		tmp = sqrt((((((l * l) * n) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = sqrt(Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(U) * sqrt(Float64(n * t))) * Float64(-1.0 * sqrt(2.0))));
	elseif (t_1 <= Inf)
		tmp = sqrt(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) - Float64(n * Float64(Float64(Float64(l / Om) * Float64(l / Om)) * Float64(U - U_42_)))) * Float64(Float64(n + n) * U)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[Sqrt[N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(-1.0 * N[(N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[Sqrt[N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] - N[(n * N[(N[(N[(l / Om), $MachinePrecision] * N[(l / Om), $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   4. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{1} \cdot \sqrt{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      10. lower-sqrt.f64 36.0

          \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   6. Applied rewrites 36.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      8. lift-*.f64 21.1

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

   4. Applied rewrites 20.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* -1.0 (* (* (sqrt U) (sqrt (* n t))) (* -1.0 (sqrt 2.0))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l (/ l Om)) t)))
       (sqrt
        (*
         (* (* (* (* l l) n) (fma n (/ (- U U*) (* Om Om)) (/ 2.0 Om))) U)
         -2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * ((sqrt(U) * sqrt((n * t))) * (-1.0 * sqrt(2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt((((((l * l) * n) * fma(n, ((U - U_42_) / (Om * Om)), (2.0 / Om))) * U) * -2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(U) * sqrt(Float64(n * t))) * Float64(-1.0 * sqrt(2.0))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(l * l) * n) * fma(n, Float64(Float64(U - U_42_) / Float64(Om * Om)), Float64(2.0 / Om))) * U) * -2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision] * N[(n * N[(N[(U - U$42$), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision] + N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * U), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   4. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{1} \cdot \sqrt{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      10. lower-sqrt.f64 36.0

          \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   6. Applied rewrites 36.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      8. lift-*.f64 21.1

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om}\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{t}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)} \]

      5. pow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      8. lift-/.f64 47.0

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{\color{blue}{Om}}, t\right)} \]

   4. Applied rewrites 47.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{-2 \cdot \left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left({\ell}^{2} \cdot \left(n \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)\right) \cdot \color{blue}{-2}} \]

   4. Applied rewrites 20.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(\ell \cdot \ell\right) \cdot n\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right) \cdot U\right) \cdot -2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 53.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_1 \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* -1.0 (* (* (sqrt U) (sqrt (* n t))) (* -1.0 (sqrt 2.0))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l (/ l Om)) t)))
       (sqrt (* t_1 (- (* (* l l) (/ (+ 2.0 (/ (* n (- U U*)) Om)) Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * ((sqrt(U) * sqrt((n * t))) * (-1.0 * sqrt(2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt((t_1 * -((l * l) * ((2.0 + ((n * (U - U_42_)) / Om)) / Om))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(U) * sqrt(Float64(n * t))) * Float64(-1.0 * sqrt(2.0))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = sqrt(Float64(t_1 * Float64(-Float64(Float64(l * l) * Float64(Float64(2.0 + Float64(Float64(n * Float64(U - U_42_)) / Om)) / Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(t$95$1 * (-N[(N[(l * l), $MachinePrecision] * N[(N[(2.0 + N[(N[(n * N[(U - U$42$), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   4. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{1} \cdot \sqrt{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      10. lower-sqrt.f64 36.0

          \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   6. Applied rewrites 36.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      8. lift-*.f64 21.1

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om}\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{t}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)} \]

      5. pow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      8. lift-/.f64 47.0

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{\color{blue}{Om}}, t\right)} \]

   4. Applied rewrites 47.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-1 \cdot \left({\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\mathsf{neg}\left({\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-{\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-{\ell}^{2} \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      4. pow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \left(2 \cdot \frac{1}{Om} + \frac{n \cdot \left(U - U*\right)}{{Om}^{2}}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \left(\frac{n \cdot \left(U - U*\right)}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      7. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \left(n \cdot \frac{U - U*}{{Om}^{2}} + 2 \cdot \frac{1}{Om}\right)\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{{Om}^{2}}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      11. unpow2 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, 2 \cdot \frac{1}{Om}\right)\right)} \]

      13. associate-*r/ N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2 \cdot 1}{Om}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right)} \]

      15. lower-/.f64 20.6

          \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right)} \]

   4. Applied rewrites 20.6%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(-\left(\ell \cdot \ell\right) \cdot \mathsf{fma}\left(n, \frac{U - U*}{Om \cdot Om}, \frac{2}{Om}\right)\right)}} \]

   5. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]

      5. lift--.f64 21.8

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]

   7. Applied rewrites 21.8%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-\left(\ell \cdot \ell\right) \cdot \frac{2 + \frac{n \cdot \left(U - U*\right)}{Om}}{Om}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 52.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\sqrt{t\_1 \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*)))))))
   (if (<= t_2 0.0)
     (* -1.0 (* (* (sqrt U) (sqrt (* n t))) (* -1.0 (sqrt 2.0))))
     (if (<= t_2 INFINITY)
       (sqrt (* t_1 (fma -2.0 (* l (/ l Om)) t)))
       (sqrt (* (+ n n) (* U (/ (* U* (* (* l l) n)) (* Om Om)))))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * ((sqrt(U) * sqrt((n * t))) * (-1.0 * sqrt(2.0)));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * fma(-2.0, (l * (l / Om)), t)));
	} else {
		tmp = sqrt(((n + n) * (U * ((U_42_ * ((l * l) * n)) / (Om * Om)))));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(U) * sqrt(Float64(n * t))) * Float64(-1.0 * sqrt(2.0))));
	elseif (t_2 <= Inf)
		tmp = sqrt(Float64(t_1 * fma(-2.0, Float64(l * Float64(l / Om)), t)));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * Float64(Float64(U_42_ * Float64(Float64(l * l) * n)) / Float64(Om * Om)))));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[Sqrt[N[(t$95$1 * N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * N[(N[(U$42$ * N[(N[(l * l), $MachinePrecision] * n), $MachinePrecision]), $MachinePrecision] / N[(Om * Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   4. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{1} \cdot \sqrt{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      10. lower-sqrt.f64 36.0

          \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   6. Applied rewrites 36.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      8. lift-*.f64 21.1

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < +inf.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om}\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{t}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)} \]

      5. pow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      8. lift-/.f64 47.0

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{\color{blue}{Om}}, t\right)} \]

   4. Applied rewrites 47.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}} \]

   if +inf.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in U* around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}}\right)} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{\color{blue}{{Om}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{\color{blue}{Om}}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left({\ell}^{2} \cdot n\right)}{{Om}^{2}}\right)} \]

      4. pow2 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{{Om}^{2}}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{{Om}^{2}}\right)} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot \color{blue}{Om}}\right)} \]

      7. lift-*.f64 16.4

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot \color{blue}{Om}}\right)} \]

   6. Applied rewrites 16.4%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{\frac{U* \cdot \left(\left(\ell \cdot \ell\right) \cdot n\right)}{Om \cdot Om}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(2 \cdot n\right) \cdot U\\ t_2 := \sqrt{t\_1 \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ t_3 := \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;-1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 10^{+133}:\\ \;\;\;\;\sqrt{t\_1 \cdot t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(t\_3 \cdot n\right) \cdot U\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1 (* (* 2.0 n) U))
        (t_2
         (sqrt
          (*
           t_1
           (-
            (- t (* 2.0 (/ (* l l) Om)))
            (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
        (t_3 (fma -2.0 (* l (/ l Om)) t)))
   (if (<= t_2 0.0)
     (* -1.0 (* (* (sqrt U) (sqrt (* n t))) (* -1.0 (sqrt 2.0))))
     (if (<= t_2 1e+133) (sqrt (* t_1 t_3)) (sqrt (* (* (* t_3 n) U) 2.0))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = (2.0 * n) * U;
	double t_2 = sqrt((t_1 * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)))));
	double t_3 = fma(-2.0, (l * (l / Om)), t);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = -1.0 * ((sqrt(U) * sqrt((n * t))) * (-1.0 * sqrt(2.0)));
	} else if (t_2 <= 1e+133) {
		tmp = sqrt((t_1 * t_3));
	} else {
		tmp = sqrt((((t_3 * n) * U) * 2.0));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(2.0 * n) * U)
	t_2 = sqrt(Float64(t_1 * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_)))))
	t_3 = fma(-2.0, Float64(l * Float64(l / Om)), t)
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(-1.0 * Float64(Float64(sqrt(U) * sqrt(Float64(n * t))) * Float64(-1.0 * sqrt(2.0))));
	elseif (t_2 <= 1e+133)
		tmp = sqrt(Float64(t_1 * t_3));
	else
		tmp = sqrt(Float64(Float64(Float64(t_3 * n) * U) * 2.0));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(t$95$1 * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(-1.0 * N[(N[(N[Sqrt[U], $MachinePrecision] * N[Sqrt[N[(n * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+133], N[Sqrt[N[(t$95$1 * t$95$3), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(t$95$3 * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 0.0

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   4. Taylor expanded in t around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot \sqrt{2}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\color{blue}{\left(\sqrt{-1}\right)}}^{2} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{\left(\frac{2}{2}\right)} \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left({-1}^{1} \cdot \sqrt{2}\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{\color{blue}{2}}\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \color{blue}{\sqrt{2}}\right)\right) \]

      10. lower-sqrt.f64 36.0

          \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   6. Applied rewrites 36.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right)} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\sqrt{U \cdot \left(n \cdot t\right)} \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      4. sqrt-prod N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

      6. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      7. lower-sqrt.f64 N/A

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

      8. lift-*.f64 21.1

         \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(-1 \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 21.1%

      \[\leadsto -1 \cdot \left(\left(\sqrt{U} \cdot \sqrt{n \cdot t}\right) \cdot \left(\color{blue}{-1} \cdot \sqrt{2}\right)\right) \]

   if 0.0 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))) < 1e133

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)}} \]
   3. Step-by-step derivation

      1. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + \color{blue}{\left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(t + -2 \cdot \frac{\color{blue}{{\ell}^{2}}}{Om}\right)} \]

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(-2 \cdot \frac{{\ell}^{2}}{Om} + \color{blue}{t}\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \color{blue}{\frac{{\ell}^{2}}{Om}}, t\right)} \]

      5. pow2 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right)} \]

      6. associate-/l* N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \color{blue}{\frac{\ell}{Om}}, t\right)} \]

      8. lift-/.f64 47.0

         \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{\color{blue}{Om}}, t\right)} \]

   4. Applied rewrites 47.0%

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right)}} \]

   if 1e133 < (sqrt.f64 (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      11. pow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      14. lift-/.f64 47.6

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 47.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 48.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{+105}:\\ \;\;\;\;\sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 1.95e+105)
   (sqrt (* (* (* (fma -2.0 (* l (/ l Om)) t) n) U) 2.0))
   (* (sqrt t) (sqrt (* (+ n n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 1.95e+105) {
		tmp = sqrt((((fma(-2.0, (l * (l / Om)), t) * n) * U) * 2.0));
	} else {
		tmp = sqrt(t) * sqrt(((n + n) * U));
	}
	return tmp;
}

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 1.95e+105)
		tmp = sqrt(Float64(Float64(Float64(fma(-2.0, Float64(l * Float64(l / Om)), t) * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 1.95e+105], N[Sqrt[N[(N[(N[(N[(-2.0 * N[(l * N[(l / Om), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999989e105

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in n around 0

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot \left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right)\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(\left(t - 2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      7. fp-cancel-sub-sign-inv N/A

         \[\leadsto \sqrt{\left(\left(\left(t + \left(\mathsf{neg}\left(2\right)\right) \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      8. metadata-eval N/A

         \[\leadsto \sqrt{\left(\left(\left(t + -2 \cdot \frac{{\ell}^{2}}{Om}\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\left(\left(\left(-2 \cdot \frac{{\ell}^{2}}{Om} + t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{{\ell}^{2}}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      11. pow2 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \frac{\ell \cdot \ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      12. associate-/l* N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      13. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

      14. lift-/.f64 47.6

          \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 47.6%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 1.94999999999999989e105 < t

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(n + n\right) \cdot U}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{\left(n + n\right) \cdot U} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 21.7

         \[\leadsto \sqrt{t} \cdot \sqrt{\left(n + n\right) \cdot U} \]

   8. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{\left(n + n\right) \cdot U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 41.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+302}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (let* ((t_1
         (*
          (* (* 2.0 n) U)
          (-
           (- t (* 2.0 (/ (* l l) Om)))
           (* (* n (pow (/ l Om) 2.0)) (- U U*))))))
   (if (<= t_1 5e+203)
     (sqrt (* (+ n n) (* U t)))
     (if (<= t_1 2e+302)
       (sqrt (* (* (* t n) U) 2.0))
       (* (sqrt (* U* U)) (/ (* (* (sqrt 2.0) n) l) Om))))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e+203) {
		tmp = sqrt(((n + n) * (U * t)));
	} else if (t_1 <= 2e+302) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((2.0d0 * n) * u) * ((t - (2.0d0 * ((l * l) / om))) - ((n * ((l / om) ** 2.0d0)) * (u - u_42)))
    if (t_1 <= 5d+203) then
        tmp = sqrt(((n + n) * (u * t)))
    else if (t_1 <= 2d+302) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((u_42 * u)) * (((sqrt(2.0d0) * n) * l) / om)
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * Math.pow((l / Om), 2.0)) * (U - U_42_)));
	double tmp;
	if (t_1 <= 5e+203) {
		tmp = Math.sqrt(((n + n) * (U * t)));
	} else if (t_1 <= 2e+302) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((U_42_ * U)) * (((Math.sqrt(2.0) * n) * l) / Om);
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * math.pow((l / Om), 2.0)) * (U - U_42_)))
	tmp = 0
	if t_1 <= 5e+203:
		tmp = math.sqrt(((n + n) * (U * t)))
	elif t_1 <= 2e+302:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((U_42_ * U)) * (((math.sqrt(2.0) * n) * l) / Om)
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	t_1 = Float64(Float64(Float64(2.0 * n) * U) * Float64(Float64(t - Float64(2.0 * Float64(Float64(l * l) / Om))) - Float64(Float64(n * (Float64(l / Om) ^ 2.0)) * Float64(U - U_42_))))
	tmp = 0.0
	if (t_1 <= 5e+203)
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(U_42_ * U)) * Float64(Float64(Float64(sqrt(2.0) * n) * l) / Om));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	t_1 = ((2.0 * n) * U) * ((t - (2.0 * ((l * l) / Om))) - ((n * ((l / Om) ^ 2.0)) * (U - U_42_)));
	tmp = 0.0;
	if (t_1 <= 5e+203)
		tmp = sqrt(((n + n) * (U * t)));
	elseif (t_1 <= 2e+302)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((U_42_ * U)) * (((sqrt(2.0) * n) * l) / Om);
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := Block[{t$95$1 = N[(N[(N[(2.0 * n), $MachinePrecision] * U), $MachinePrecision] * N[(N[(t - N[(2.0 * N[(N[(l * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * N[Power[N[(l / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(U - U$42$), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+203], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$1, 2e+302], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(U$42$ * U), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * n), $MachinePrecision] * l), $MachinePrecision] / Om), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 4.99999999999999994e203

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.7

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]

   if 4.99999999999999994e203 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*)))) < 2.0000000000000002e302

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 36.1

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 2.0000000000000002e302 < (*.f64 (*.f64 (*.f64 #s(literal 2 binary64) n) U) (-.f64 (-.f64 t (*.f64 #s(literal 2 binary64) (/.f64 (*.f64 l l) Om))) (*.f64 (*.f64 n (pow.f64 (/.f64 l Om) #s(literal 2 binary64))) (-.f64 U U*))))

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in U* around inf

      \[\leadsto \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om} \cdot \sqrt{U \cdot U*}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \color{blue}{\frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{Om}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{U \cdot U*} \cdot \frac{\color{blue}{\ell \cdot \left(n \cdot \sqrt{2}\right)}}{Om} \]

      4. *-commutative N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\color{blue}{\ell} \cdot \left(n \cdot \sqrt{2}\right)}{Om} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\color{blue}{\ell} \cdot \left(n \cdot \sqrt{2}\right)}{Om} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\ell \cdot \left(n \cdot \sqrt{2}\right)}{\color{blue}{Om}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\left(n \cdot \sqrt{2}\right) \cdot \ell}{Om} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\left(n \cdot \sqrt{2}\right) \cdot \ell}{Om} \]

      9. *-commutative N/A

         \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \]

      11. lower-sqrt.f64 14.3

          \[\leadsto \sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om} \]

   4. Applied rewrites 14.3%

      \[\leadsto \color{blue}{\sqrt{U* \cdot U} \cdot \frac{\left(\sqrt{2} \cdot n\right) \cdot \ell}{Om}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 38.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 3 \cdot 10^{-287}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n + n} \cdot \sqrt{U \cdot t}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= n 3e-287)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt (+ n n)) (sqrt (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3e-287) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt((n + n)) * sqrt((U * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (n <= 3d-287) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt((n + n)) * sqrt((u * t))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (n <= 3e-287) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt((n + n)) * Math.sqrt((U * t));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if n <= 3e-287:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt((n + n)) * math.sqrt((U * t))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (n <= 3e-287)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(Float64(n + n)) * sqrt(Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (n <= 3e-287)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt((n + n)) * sqrt((U * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[n, 3e-287], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(n + n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(U * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.99999999999999992e-287

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 36.1

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 2.99999999999999992e-287 < n

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.7

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   7. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot t\right)}} \]

      3. sqrt-prod N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot t}} \]

      4. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot t} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot t}} \]

      6. lift-+.f64 N/A

         \[\leadsto \sqrt{\color{blue}{n + n}} \cdot \sqrt{U \cdot t} \]

      7. lift-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{n + n}} \cdot \sqrt{U \cdot t} \]

      8. lower-sqrt.f64 20.8

         \[\leadsto \sqrt{n + n} \cdot \color{blue}{\sqrt{U \cdot t}} \]

   8. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\sqrt{n + n} \cdot \sqrt{U \cdot t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 37.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t} \cdot \sqrt{\left(n + n\right) \cdot U}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= t 2e+134)
   (sqrt (* (* (* t n) U) 2.0))
   (* (sqrt t) (sqrt (* (+ n n) U)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2e+134) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(t) * sqrt(((n + n) * U));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (t <= 2d+134) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(t) * sqrt(((n + n) * u))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (t <= 2e+134) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(t) * Math.sqrt(((n + n) * U));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if t <= 2e+134:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt(t) * math.sqrt(((n + n) * U))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (t <= 2e+134)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = Float64(sqrt(t) * sqrt(Float64(Float64(n + n) * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (t <= 2e+134)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt(t) * sqrt(((n + n) * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[t, 2e+134], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[t], $MachinePrecision] * N[Sqrt[N[(N[(n + n), $MachinePrecision] * U), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999984e134

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 36.1

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 1.99999999999999984e134 < t

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 53.0%

      \[\leadsto \sqrt{\color{blue}{\left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right) \cdot \left(\left(n + n\right) \cdot U\right)}} \]

   5. Applied rewrites 25.7%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\frac{\ell \cdot \ell}{Om \cdot Om} \cdot \left(U - U*\right)\right)} \cdot \sqrt{\left(n + n\right) \cdot U}} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{\left(n + n\right) \cdot U} \]
   7. Step-by-step derivation

      1. lower-sqrt.f64 21.7

         \[\leadsto \sqrt{t} \cdot \sqrt{\left(n + n\right) \cdot U} \]

   8. Applied rewrites 21.7%

      \[\leadsto \color{blue}{\sqrt{t}} \cdot \sqrt{\left(n + n\right) \cdot U} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 36.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;Om \leq 10^{+25}:\\ \;\;\;\;\sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (n U t l Om U*)
 :precision binary64
 (if (<= Om 1e+25) (sqrt (* (* (* t n) U) 2.0)) (sqrt (* (+ n n) (* U t)))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1e+25) {
		tmp = sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = sqrt(((n + n) * (U * t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    real(8) :: tmp
    if (om <= 1d+25) then
        tmp = sqrt((((t * n) * u) * 2.0d0))
    else
        tmp = sqrt(((n + n) * (u * t)))
    end if
    code = tmp
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	double tmp;
	if (Om <= 1e+25) {
		tmp = Math.sqrt((((t * n) * U) * 2.0));
	} else {
		tmp = Math.sqrt(((n + n) * (U * t)));
	}
	return tmp;
}

Python

def code(n, U, t, l, Om, U_42_):
	tmp = 0
	if Om <= 1e+25:
		tmp = math.sqrt((((t * n) * U) * 2.0))
	else:
		tmp = math.sqrt(((n + n) * (U * t)))
	return tmp

Julia

function code(n, U, t, l, Om, U_42_)
	tmp = 0.0
	if (Om <= 1e+25)
		tmp = sqrt(Float64(Float64(Float64(t * n) * U) * 2.0));
	else
		tmp = sqrt(Float64(Float64(n + n) * Float64(U * t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(n, U, t, l, Om, U_42_)
	tmp = 0.0;
	if (Om <= 1e+25)
		tmp = sqrt((((t * n) * U) * 2.0));
	else
		tmp = sqrt(((n + n) * (U * t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := If[LessEqual[Om, 1e+25], N[Sqrt[N[(N[(N[(t * n), $MachinePrecision] * U), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if Om < 1.00000000000000009e25

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   2. Taylor expanded in t around inf

      \[\leadsto \sqrt{\color{blue}{2 \cdot \left(U \cdot \left(n \cdot t\right)\right)}} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(U \cdot \left(n \cdot t\right)\right) \cdot \color{blue}{2}} \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\left(n \cdot t\right) \cdot U\right) \cdot 2} \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

      6. lower-*.f64 36.1

         \[\leadsto \sqrt{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2} \]

   4. Applied rewrites 36.1%

      \[\leadsto \sqrt{\color{blue}{\left(\left(t \cdot n\right) \cdot U\right) \cdot 2}} \]

   if 1.00000000000000009e25 < Om

   1. Initial program 50.5%

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

   3. Applied rewrites 52.8%

      \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

   4. Taylor expanded in t around inf

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
   5. Step-by-step derivation

      1. lower-*.f64 35.7

         \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

   6. Applied rewrites 35.7%

      \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 35.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(n + n\right) \cdot \left(U \cdot t\right)} \end{array} \]

FPCore

(FPCore (n U t l Om U*) :precision binary64 (sqrt (* (+ n n) (* U t))))

C

double code(double n, double U, double t, double l, double Om, double U_42_) {
	return sqrt(((n + n) * (U * t)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(n, u, t, l, om, u_42)
use fmin_fmax_functions
    real(8), intent (in) :: n
    real(8), intent (in) :: u
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: u_42
    code = sqrt(((n + n) * (u * t)))
end function

Java

public static double code(double n, double U, double t, double l, double Om, double U_42_) {
	return Math.sqrt(((n + n) * (U * t)));
}

Python

def code(n, U, t, l, Om, U_42_):
	return math.sqrt(((n + n) * (U * t)))

Julia

function code(n, U, t, l, Om, U_42_)
	return sqrt(Float64(Float64(n + n) * Float64(U * t)))
end

MATLAB

function tmp = code(n, U, t, l, Om, U_42_)
	tmp = sqrt(((n + n) * (U * t)));
end

Wolfram

code[n_, U_, t_, l_, Om_, U$42$_] := N[Sqrt[N[(N[(n + n), $MachinePrecision] * N[(U * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 50.5%

   \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)} \]

3. Applied rewrites 52.8%

   \[\leadsto \sqrt{\color{blue}{\left(n + n\right) \cdot \left(U \cdot \left(\mathsf{fma}\left(-2, \ell \cdot \frac{\ell}{Om}, t\right) - n \cdot \left(\left(\frac{\ell}{Om} \cdot \frac{\ell}{Om}\right) \cdot \left(U - U*\right)\right)\right)\right)}} \]

4. Taylor expanded in t around inf

   \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
5. Step-by-step derivation

   1. lower-*.f64 35.7

      \[\leadsto \sqrt{\left(n + n\right) \cdot \left(U \cdot \color{blue}{t}\right)} \]

6. Applied rewrites 35.7%

   \[\leadsto \sqrt{\left(n + n\right) \cdot \color{blue}{\left(U \cdot t\right)}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2.0 n) U) (- (- t (* 2.0 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2.0)) (- U U*))))))